Chapter 4: Evolution Equations

This chapter develops the equations that describe how destiny states change over time, providing the mathematical machinery for calculating probability evolution.

4.1 The Destiny Schrödinger Equation

The fundamental equation of destiny evolution is:

$$i\hbar_{destiny} \frac{\partial}{\partial t}|\Psi(t)\rangle = \hat{H}_{destiny}|\Psi(t)\rangle$$

This equation describes how the destiny state vector |Ψ(t)⟩ changes continuously in time under the influence of the Destiny Hamiltonian.

4.2 Hamiltonian Components

The Destiny Hamiltonian decomposes into four components:

$$\hat{H}_{destiny} = \hat{H}_{transit} + \hat{H}_{stage} + \hat{H}_{choice} + \hat{H}_{env}$$

Component	Physical Meaning	Time Dependence
$\hat{H}_{transit}$	Planetary transits and cosmic cycles	Periodic
$\hat{H}_{stage}$	Life stage transitions (childhood, adolescence, etc.)	Step functions
$\hat{H}_{choice}$	Free will and conscious decisions	Stochastic
$\hat{H}_{env}$	Environmental and social factors	Variable

4.3 Time Evolution Operator

The formal solution to the Destiny Schrödinger Equation is:

$$|\Psi(t)\rangle = \hat{U}(t, t_0)|\Psi(t_0)\rangle$$

Where the time evolution operator is:

$$\hat{U}(t, t_0) = \mathcal{T} \exp\left(-\frac{i}{\hbar_{destiny}}\int_{t_0}^{t} \hat{H}(t') dt'\right)$$

Here 𝒯 denotes time-ordering, necessary because the Hamiltonian may not commute with itself at different times.

4.4 Transition Probabilities

The probability of transitioning from state |i⟩ to state |f⟩ over time interval [t_0, t] is:

$$P(i \rightarrow f) = |\langle f | \hat{U}(t, t_0) | i \rangle|^2$$

This formula is the basis for all predictive calculations in MetaDestiny.